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O'BRIEN'S
MATHEMATICAL TRACTS,
PART I.
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I 1. -
MATHEMATICAL TRACTS,
PART I.
MATHEMATICAL TRACTS,
PART I.
LAPLACE'S COEFFICIENTS,
THE FIGURE OF THE EARTH,
THE MOTION OF A RIGID BODY ABOUT ITS
CENTER OF GRAVITY,
AND
PRECESSION AND NUTATION.
MATTHEW O'BRIEN, B.A.,
•MATHEMATICAL LECTURER OP CAIUS COLLEGE.
CAMBRIDGE:
PRINTED AT THE UNIVERSITY PRESS,
FOR J. & J. J. DEIGHTON, TRINITY STREET;
AND
JOHN W. PARKER, LONDON.
M.DCCC.XL.
Stack
Annex
233
PREFACE.
THE subjects treated of in the following Tracts
are, Laplace's Coefficients; the Investigation of the
Figure of the Earth on the Hypothesis of its Original
Fluidity ; the Equations of Motion of a Rigid Body
about its Center of Gravity; and the Application of
these Equations to the case of the Earth. The
first of these subjects should be familiar to every
Mathematical Student, both for its own sake, and
also on account of the many branches of Physical
Science to which it is applicable. The second sub-
ject is extremely interesting as a physical theory,
bearing upon the original state of the Earth and
of the planetary bodies; it is also well worthy of
attention on account of the important and exten-
sive observations which have been made in order
to verify it. The Author has put both these sub-
jects together, commencing with the Figure of the
Earth, and introducing Laplace's Coefficients when
occasion required them; this being perhaps the best
VI PREFACE.
and simplest way of exhibiting the nature and use
of these coefficients.
The Author has treated some parts of these sub-
jects differently from the manner in which they
are usually treated, and he hopes that by so doing
he has avoided some intricate reasoning and trou-
blesome calculation, and made the whole more
accessible to students of moderate mathematical at-
tainments than it has hitherto been.
In calculating the attractions of the Earth on
any particle, he has arrived at the correct results,
without considering diverging series as inadmissible ;
and this he conceives to be important, because there
is evidently no good reason why a diverging series
should not be as good a symbolical representative of a
quantity as a converging series ; or why there should
be any occasion to enquire whether a series is di-
verging or converging, as long as we do not want
to calculate its arithmetical value or determine its
sign. Instances, it is true, have been brought for-
wprd by Poisson* in which the use of diverging
series appears to lead to error; but if the reason-
ing employed in Chapter in. of these Tracts be not
incorrect, this error is due to quite a different cause ;
• See Bowditch's Laplace. Vol. H. p. 167.
PREFACE. Til
as will be immediately perceived on referring to Ar-
ticles 33, 34, 35, and 37.
The Author has deduced the equations of motion
of a rigid body about its center of gravity by a
method which he hopes will be found, less objec-
tionable than that in which the composition and re-
solution of angular velocities are employed, and less
complex than that given by Laplace and Poisson ;
he has also endeavoured to simplify the application
of these equations to the case of the Earth.
In the First Part of these Tracts he has confined
himself to the most prominent and important parts of
each subject. In the Second Part, which will shortly
be published, he intends, among other- things, to give
some account of the controversies which Laplace's Co-
efficients have given rise to ; to investigate more fully
the nature and properties of these functions ; to give
instances of their use in various problems ; for this
purpose to explain the mathematical theory of Elec-
tricity ; to consider more particularly the Equations
of motion of a rigid body about its center of gra-
vity, and the conclusions that may be drawn from
them ; to give the theory of Jupiter's Satellites, and
of Librations of the Moon ; 'and to say something
on the subject of Tides.
Vlll PREFACE.
The Author has not given the investigation of
the effect of the Earth's Oblateness on the motions
of the Moon, but he has endeavoured to prove that
this effect does not afford any additional evidence
of the Earth's original Fluidity beyond that which
may be obtained from the Figure of the Earth, and
Law of Gravity.
MATHEMATICAL TRACTS,
PART I.
CHAPTER I.
FIGURE OF THE EAUTH.
1. IT has been well ascertained, by extensive and
accurate geodetical measurements, that the general figure of
the Earth is that of an oblate surface of revolution, de-
scribed about the axis of diurnal rotation : and this fact
suggests the idea, that the diurnal rotation may be in some
way or other the cause of this peculiar figure, especially
if we consider that the Sun and planets, which all rotate
like the Earth, appear also to have the same sort of oblate
form of revolution about their axes of rotation.
The most obvious and natural way of accounting for
the influence thus apparently exerted on the figures of
the planetary bodies by their rotation, is to suppose that
they may once have been in a state of fluidity ; for, con-
ceive a fluid gravitating mass to be gradually put into a
state of rotation round a fixed axis : it is evident that be-
fore the motion commenced it would, according to a well-
known hydrostatical law, be arranged all through in con-
centric spherical strata of equal density ; but on the motion
of rotation commencing a centrifugal force would arise,
which would be greater at greater distances from the axis,
and would therefore evidently produce an oblateness in the
forms of the strata, leaving them still symmetrical with
respect to the axis. Thus the hypothesis of the original
fluidity of the bodies of our system, considered in connection
with their rotation, accounts for their oblate form.
1
2
2. To account for the present solidity of the surface
of our own planet, we may suppose that its temperature was
originally so great as to keep it in a state of fusion, and
that this was the cause of its fluidity ; but that, in the
course of ages, it, at least its surface, has cooled down and
hardened into its present consistence. This supposition is
borne out by geological facts ; and it is by no means un-
likely, if we consider that the principal body of our system
is at present most probably in a state of fusion.
3. This hypothesis of the Earth's original fluidity re-
ceives much confirmation from observations on the intensity
and direction of the force of gravity ; for it follows from
the hydrostatical law already alluded to, that the Earth, if
fluid, ought to consist entirely of equidense strata of the
same sort of form as the exterior surface*, and therefore
the whole mass ought to be arranged symmetrically with
respect to the axis of rotation, and nearly so with respect to
the centre of that axis. Hence, the force of gravity, which
is the resultant of the Earth's attraction and the centrifugal
force, ought to be the same at all places in the same latitude,
and nearly the same at all places in the same meridian.
Moreover it follows from another hydrostatical law, that
the direction of this force of gravity ought to be every
where perpendicular to the surface.
Now all this has been proved to be the case by nume-
rous observations with pendulums, plumb-lines, levels, &c.
(omitting very small variations, which may be easily ac-
counted for in most cases). Hence, the hypothesis of the
Earth's original fluidity is confirmed by the observations
which have been made on the force of gravity.
4. But this hypothesis has been advanced almost to a
moral certainty, by investigating precisely what effect it
ought to have, if true, on the arrangement of the Earth's
* We suppose the Earth to be heterogeneous, because the pressure of the
superincumbent mass must condense the central parts more than the superh'cial ;
besides, the well-known fact of the mean density of the whole Earth being greater
than the density of the superficial parts, proves that the Earth is not homo-
geneous.
mass, and by comparing the result with observation; for it is
found that if the hypothesis be true, the strata which com-
pose the Earth ought to have not only an oblate form, but
one very peculiar kind of oblate form ; and it is found that
this result admits of most satisfactory comparison with ac-
curate and varied observation, and actually coincides with
it in a most remarkable manner; from which we may con-
clude, almost with certainty, that the hypothesis is correct ;
for it is extremely difficult to account in any other way for
so marked an agreement with observation of such a very
peculiar result.
5. The object of the following pages is to give an
account of this interesting investigation, and to state briefly
the manner in which its result may be tested by observation.
In the first place, we shall determine the law of arrange-
ment of the Earth's mass, on the hypothesis of its original
fluidity, by means of Laplace's powerful and beautiful Ana-
lysis; and in the next place, we shall deduce such results
as shall admit of immediate comparison with observation.
The most important of these results are ; The expression
for the length of a meridian arc corresponding to a given
difference of latitude, and. The law of variation of the
force of gravity at different points of the Earths surface.
The other results which we shall deduce depend on certain
assumptions respecting the law of density of the Earth, and
are therefore not so important. We now proceed, in the
first place, to determine the law of arrangement of the
Earth's mass, as follows.
6. A heterogeneous jluid mass composed of par-
ticles which attract each other inversely as the square of
the distance rotates uniformly in relative equilibrium*
round a fixed axis : to determine the law of its arrange-
ment.
Take the axis of rotation as that of #, and let xyz
be the co-ordinates of any particle $m, XYZ the resolved
* By relative equilibrium we mean that the particles of the mass, though
actually moving, are at rest relatively to each other.
attractions ol' the mass on $w, p and p the density and
pressure at the point (#ysf), and o> the angular velocity
of the mass. Then, by the principles of Hydrostatics,
we have
dp = p \Xdx + Ydy + Zdz + w8 (xdx + ydy)} ... (l)
To calculate the expression (Xdcc + Ydy + Zdz), let $m
be any attracting particle, and x'y'% its co-ordinates ; then
we have
and similar expressions for Y and Z.
Now assume (F) to denote the expression
VV- *)2 + (y- y?+ (*' - *)2 '
i. e. the sum of each particle divided by its distance from
§m. Then it is evident that
y dV dV dV
= Tx* ~~dy* ~dx'
and Xdfc + Ydy + Zdz = —— dx + —:—dy -\ -- dx.
dx dy ' dss
and therefore the equation (1) becomes
The coefficient of p here is a complete differential ;
hence by the principles of Hydrostatics, the necessary
and sufficient conditions of equilibrium are, that the whole
mass be arranged in strata of equal density, the general
equation to any one of them being
C being a constant different for different strata, the
exterior surface being one of these strata, since it is a
free surface.
7- Hence the equations from which the problem is
to be solved are
(A).
=
8. These equations are unfortunately very much in-
volved in each other, so much so as to be scarcely manage-
able ; for V must be found by integration between limits
which depend on the form of the exterior stratum, and
therefore on the equation (A) ; and also the law of density,
and therefore the form of the internal strata, and therefore
the equation (A), must be known in order to calculate F.
But V is itself involved in (A), hence (A) cannot be
made use of in calculating V. It will therefore be neces-
sary to devise some way of eliminating F, without knowing
what function it is. To do this in the general case is be-
yond the present powers of analysis ; but in the particular
case we are concerned with, the fact of the strata being all
nearly spherical, introduces considerable simplification, and
by using the ingenious analysis due to Laplace, we shall
be able to eliminate V with comparative ease, at least, ap-
proximately, but with quite sufficient accuracy.
9. In the first place, the strata being nearly spherical,
we shall find it convenient to make use of polar instead of
rectangular co-ordinates, and we shall accordingly transform
our equations as follows :
Let r, 0, (f>, /, 0', 0', be the co-ordinates of $m and $m'
respectively; r, 0, 0, signifying the same as in Hymers1
Geometry of three dimensions, page (77). Then we have
off = r sin 9 cos d),
y = r sin 0 sin <^>,
% = r cos 9,
and similar expressions for v't y', %'.
Hence the equation (A) becomes
C = V + — r- sin2 9,
2
and the equation (/?) becomes, observing that
$m = p r'2 sin 9' dr d& dfi
c^ r
Jo Jo
\Xr'-2rr'|cos#cos#'+sin0sin0'cos(0-^)')}-fr'2
0 and r, being the limits of r', 0 and TT of 9', and 0 and
2?r of 0', 7*1 being the value of r at the surface, and there-
fore in general a function of 0' and <p'. It will of course
on this account be necessary to integrate first with respect
to /, but it is no matter in what order we perform the
integrations relative to 0' and <p', since their limits are
constants.
If these integrations could be performed, V would come
out a function of r, 0, and <^>, and unknown constants de-
pending on the form of the strata and the law of density.
10. We shall find it convenient to put /* and /u? for
cos 6 and cos 9' respectively, this will give
sin 9'd9' = - dp.',
and the limits of /u. will be — 1 and 1 ; or we may put dfi
instead of — d//, if at the same time we reverse the limits
of fji. Hence our equations become
(A'),
When for brevity we have put
cos 9 cos ff + sin 9 sin 9' cos (0 - <^>') = p,
i. c. fifj! + \/l -fj? . \A - //* • cos (^> ~ 0') = P-
11. We shall now introduce into these equations the
condition that the strata are nearly spherical. If the strata
were actually spherical, the whole mass would be symmetri-
cal with respect to the centre, and therefore V being the
sum of each particle divided by its distance from $m would
depend simply on the distance of $m from the centre, and
therefore be the same at all points of the same stratum.
We may hence conclude, that if the strata instead of being
actually spherical be only nearly so, V also, though not
actually the same, will yet be nearly the same at all points
of the same stratum. Now the value of V for any stratum
is given by the equation {A') i. e.
but V (as we have shewn) ought to be nearly constant at
all points of this stratum, hence the variable part of it, viz.
— r2(l — /u2) must be always small : therefore since r2(l — fj?)
is not always small, wz must be so. We shall accordingly
take ft>2 as the standard small quantity in our approxima-
tions, neglecting its square and higher powers in the first
approximation.
12. Now to8 being a small quantity, we may suppose
the equation to any nearly spherical surface, and therefore to
any of the strata, to be put in the form
where a is the radius of any sphere which nearly coincides
with the stratum (that sphere, suppose, which includes the
same volume as the stratum*), and aa?u is the small quan-
tity to be added to a to make it equal to r, and therefore u
in general will be some function of /u. and 0. Moreover, u
will be a function of a also, otherwise the strata would be
all similar surfaces, which of course we have no right to
assume them to be ; a may be considered as the variable
* We make this supposition, at present,"for the sake of giving a definite idea
of what a is ; hereafter it will be found an advantageous supposition.
8
parameter of the system of surfaces which the strata con-
stitute. We shall introduce the variable a into our' equa-
tions instead of r, and get every thing in terms of a, /x
and <£, instead of r, /u, and 0; the advantages of this change
will soon be perceived.
13. First, then, in the equation (^'), putting a (l+a>2w)
instead, of r, and neglecting the squares and higher powers
of <*r, we find
Next we shall make a similar substitution in the equa-
tion (B') by putting r' = a (l + o>2w'), when u denotes
what u becomes when a, /*, and 0 are exchanged for a', /,
and <p' respectively.
Assume for brevity,
then in the equation (B') we shall have
* = /oai p/ (r, /) . da',
where Oj is the parameter of the exterior surface.
and substituting for r,f(r,r')=tf{r,(a'+a'u?u)}
dr 9 d(a'u)
and hence, since — 7 = 1 + to . — - — H^ , we have
da da
9
Hence, neglecting the squares and higher powers of
ft)2, the equation (B7) becomes
14. This expression for V is much more manageable
than that in the equation (B1) ; for in the first place the
limits are all constant, and therefore we may take them in
any order we please; and in the second place, p, instead
of depending on all the variables, as of course it did before,
is now a function of one variable only, viz. a \ for each
stratum being equidense throughout, it is evident that p is
the same at every point of the same stratum, and therefore
varies only when we pass from stratum to stratum, i. e. it
varies with a alone.
Hence, changing the order of the integrations, and
bringing p outside the integral signs relative to ^ and 0',
we have
~ *r a'u' d* <*' da''
15. We have thus introduced into our equations the
condition of the strata being nearly spherical.
We shall find it convenient to make a farther substitu-
tion in this equation, viz. by putting r—a(l + <o2w), which,
neglecting the squares and higher powers of <o2, gives
r P {r V-! /(a' a/> dfi' w
— ^ f /(a, a') dp.' d<b'
aaJ<\ J -\"
'an (a' ir f f(a* a'^u> dfjL d<p} \ da''
10
16. The next thing we shall do is to perform the in-
tegrations relative to /*' and 0' ; to do this we shall expand
the quantity^y^aa'), which, since it represents
a'z
\/a/2-2aa'p+a2'
a a
may be expanded in a series of powers either of — or — .
a a
The coefficients will evidently be the same whether we ex-
pand it in powers of —. or of — , they will in fact be the
a a
coefficients of the powers of h in the expansion of the
quantity
1
We shall assume Qy, Qn Q2, &c. to denote these co-
efficients, i. e. we shall assume
Q0 will evidently be unity. The rest of these coefficients
will be rational and integral functions of p, i.e. of
fifi + V 1 — yU2 . V 1 — //2 . COS (0 — 0').
It is evident that they all become unity when p becomes
unity ; for then
becomes , or 1 + h + h2 + &c.
We shall have no occasion however to determine their
forms. They (and other functions of the same character)
are the celebrated coefficients of Laplace ; they possess very
remarkable properties, which we shall now digress to inves-
tigate, as they wonderfully facilitate the integrations we
have to perform, and enable us to eliminate V from the
equations (A") and (5"'), with great facility, without know-
ing its form.
CHAPTER II.
LAPLACE S COEFFICIENTS.
17. IN order to investigate the properties of the
functions Q,, Qn Q2, SEC. introduced in the last chapter, we
shall recur to the expression from which they were origin-
ally derived, viz.
1
We meet with this expression constantly in physical
problems, especially those in which attractions are con-
cerned, and it is therefore worthy of particular consi-
deration.
Assuming R to denote this expression, we have
R
'- x? + (yf- yf + (x'-)*'
and differentiating this equation twice relatively to xyx
respectively,
dR __ (x-.v)
dx ~ K*'-*)2+(y'-y)2 + (*'-*)2}*
-JP (*'-*)
« .JP'V-.)-*
dx* dx
and similarly
fp Tt
—
12
\
hence evidently
d-JR d*R d*R _ 1
* =0 (1).
18. We shall express this differential equation in
terms of the polar co-ordinates r0(p instead of xyx. To
facilitate the transformation we shall assume an auxiliary
quantity s, such that
s = r sin 0 ;
and therefore since x = r sin 0 cos 0, and y = r sin 6 sin 0,
we shall have
X = S COS 0,
y = s sin 0.
Then considering s and 0 as independent variables
instead of x and y, we have
dR _ dR dx dR dy
ds dx ds dy ds
dR dR .
= — — cos 0 + — — sm0 (2).
dx dy
dzR d*R dzR d?R
and — — = — — cos^0+2 • — — cos 0 sin 0+ — — sm20...(3).
ds* dx2 dxdy dy2
. dR dR dR
and -**,= - — * sin d> H s cosd> (4).
(fm dx dy
d*R d*R . d*R d?R
-r = -T-r*f Sin 0 — 2 — — . S^COS 0 Sin 0 H S2 COSZ(b
d0 dx2 dxdy dy2
dR ' dR
Equations (3) and (5) give
d?R l d*R d?R tfR 1 (dR dR
~TT + 1 T3T = T~T + ~T~9 --- — cos 0 + — — sin 0
ds* s2 d02 dx2 dy2 s \dx dy
d*R d?R 1 dR
by equation (2).
13
Now we have
x = r cos 9
s — r sin 0*,
and these equations connect % s r 9 in exactly the same way
in which xys(f> are connected by the equations
X = S COS 0
y = s sin 0.
Hence we may prove exactly as before, that
dR l (PR _ d'R d*R _ i dR
d^Jr^i~d¥= "d**" + rf*2 ~ r~dr'"
adding this to the equation (6), we have by (l),
I. tfR rffl l^ d~R l dR l dB_
? d0* +~d7+ r2 drF" « ds r~d7"
Now by (2) and (4),
dR dR cos0 dR
— sin 0 + — —2- = — - ,
ds d(f) s dy
and hence observing as before, that x s r 9 are connected
together in exactly the same way as x y s 0, we have
dR . dR cos 9 dR
hence, substituting this value of — — in equation (7), and
ds
putting r sin 9 for s, and multiplying by r2, we have
d*Jl cos 9 d_R_ I d^R. ,d*R dR_ _
J¥ + sin 9 ~d9 + shT^ dtf + r ~d7 + 'T ~dr ~ °'
l d dR\ l d*R d f dR
which is the equation (l) expressed in polar co-ordinates.
* The author finds that he has been anticipated in making this use of the
auxiliary quantity s, by the Cambridge Mathematical Journal.
14
19. Now K expressed in polar co-ordinates becomes
1
Vr -
which = —
r
/ r r*
V 1 - %P - + -72
Qn n + &C.
(See Art. 16). Hence substituting this value of R in the
equation just obtained, and putting the coefficient of r"
equal to zero, since r is indeterminate, we find imme-
diately
sin 0 dO \ d9 ] sin* 0 d(f>
which is a partial differential equation of the second order,
connecting Qn with /a. and 0 ; of course, being such, it
admits of an infinite number of solutions besides Qn.
We shall have no occasion to solve it, but "we shall
find it of use in investigating the properties of Qn. All
solutions of it which are rational and integral functions
of cos 0, sin 0, cos <f), sin <p (i. e. of /x, \/l-fj?9 cos (f),
sin 0), are called Laplace's coefficients of the nth order,
having been first brought into notice by Laplace in his
Mecanique Celeste, Liv. in.: the equation itself may
be called Laplace's equation. Why we restrict Laplace's
coefficients to be rational and integral functions of /u>
'X/l - M8> cos 0 a°d sin (f>, will appear presently.
20. We may remark here that in consequence of the
linearity of Laplace's equation, the sum of any number of
15
Laplace's coefficients of the wth order is also a Laplace's co-
efficient of the wth order.
Also any constant quantity is a Laplace's coefficient of
the order 0, for if F0 be a Laplace's coefficient of the order
0, we have
d
which equation is evidently satisfied by
F0 = any constant,
and hence any constant is a Laplace's coefficient of the
order 0.
It may easily be seen by trial that
a/i, and a \/l - /u2 cos (0 + /3)
are Laplace's coefficients of the order 1, a and /3 being
constants ; and
a (1 - M2)> a jtA \A ~ M2 cos (0 + /3), a (l - fj?) cos (2 0 + /3),
are Laplace's coefficient of the order 2, and so on. We
shall not have any occasion at present to determine the
general expression for a Laplace's coefficient of the wth order,
but, to give clear ideas, we shall just state that it may be
put in the form
A0 Mn+4} (I -fjfyM^costy + aj -I- 4,(1 -ffiMn_ 2cos(2<p + a2) ,
&c ............. +A(l-
When AQ Al SEC. ... a^ a2 ... &c. are any constants, Mn Mn_^
&c. contain rational and integral functions of /u, of the di-
mensions w, n — l, n -*2, &c. respectively*.
21. The first property we shall prove of Laplace's
coefficients is this : If Ym and Zn be any two Laplace's
coefficients of the mth and wth orders respectively, then
i I Ym Zndju. d(f) = 0, except when m = n.
Jo J -i
* We shall recur to this subject in Part n. of these Tracts.
16
For since ZM satisfies Laplace's equation, we have
n (n + 1) *" Ym Zn
Now integrating by parts, and observing that 1 - M2 = 0,
at each limit we have
and similarly,
f"v fz-j r~ dr.dz
J, Y"^d* = - J. Tp-jf**.
1 /^
* observing that Ym -•--* is the same at each limit, be-
ct(p
cause Ym and Zn are functions of sin 0 and cos 0, and not
of 0 simply ; hence substituting in (l)
n.(n + l) r*r
J0 J-\
r^ ri (/ 2,dr»
-I. L(i-^^'
In exactly the same way we may shew that
f'
-I.
* This is the reason why we have assumed Laplace's coefficients to be
functions of sin <f> and cos <7>, and not of <j> simply.
17
Hence subtracting
\n(n + 1) -m(m+ 1) } * Ym Zndnd<t>= 0 .
Now the factor n(n + l) — m(m + 1) does not = 0,
except when m = n ; hence the other factor must be zero,
hence
r* r Ymznd/uid(f>=o,
J0 J -}
except when m = n.
22. Since Q0 = l (see Art. 16), we have
f
J —
= 0, when n is greater than 0,
= r^ f
'0
= 47T.
It need scarcely be remarked that Q0, Q15 Q2, &c. possess
exactly the same properties with respect to p and 0' that
they do with respect to /j. and (p.
23. We shall now have occasion to introduce a
remarkable discontinuous function, but before we do so
we shall give a simple example of functions of this
description, in order to render our reasoning more satis-
factory to those who have not been accustomed to them.
We may easily prove, by the aid of the exponential
value of the cosine of an arc, that
*(.-f)
cos a + cos (a + /3) + cos (a + 2/3) ad inf. ... = — ;
suppose here that a = — , then we find
0
cos a + cos 3 a + cos 5 a + ... = -
2 sin a
18
hence this series is always zero, except when a= any mul-
tiple of TT, in which case sin a becomes zero, and therefore
the series becomes - ; thus though each term of this series
varies continuously with a, the series itself varies discon-
tinuously, being constantly zero, except when a passes
through any of the values 0, ± TT, ± 2 ?r, &c.. when the series
suddenly becomes -; i.e., some unknown or indeterminate
quantity. To explain the nature of this series more clearly,
we observe that
sin — | cos a 4- cos (a + /5) + SEC. } = - ^ sin f a ) ,
whatever be the values of a and /3 ; suppose a = — ;
then sin I a 1 becomes zero, whatever be the value of
a ; hence
sin a {cos a + cos 3a + cos 5a + &c. } = 0,
for all values of a.
Now as long as a is not a multiple of TT, sin a will not be
zero, and therefore
cos a + cos 3 a + ... &c.
must be zero ; but if a be any multiple of TT, then sin a will
be zero, and our equation will be satisfied quite independent-
ly of its other factor, and hence will give us no information
as to the value of that factor; hence when a is any multiple
of T, cos a + cos 3 a + &c. is some unknown or indeterminate
quantity.
It is important to remark that the change in the value of
cos a + cos 3a + &c., when a becomes a multiple of -a-, is
perfectly sudden, for since the second member of the equa-
tion is always absolutely zero, it is evident that as long as
sin a is not actually zero, though it differs from it by ever so
19
small a quantity, cos a + cos 3 a + &c ---- must be so; for
this reason cos a + cos 3 a + Sic. is called a discontinuous
function.
24. We shall now bring forward the remarkable dis-
continuous function we alluded to: it is the following series,
viz.
Qo+ 3Q/+ &c- + (2w + 1) Q«+ &c. ad inf.
this series is of exactly the same nature as that we have
just considered, being a function of the variables /x and <£»,
which is always zero, except for certain particular values of
these variables.
To shew this, we have
Q + Q,A + ... Qnh» + &c. = , ~ ,
VI - 2ph + h?
and differentiating this relatively to /*and multiplying by 2 h,
and adding these equations,
«,+ »(t*...(g. + .) «.*•+ &c. - (1--^¥
now here put h — 1 and we find
0
O, + 3 Q + &c. = - — — = ,
(2 - 2p)f
hence QQ+ 3Q7 + &c. = 0 for all values of />, except p
0
when it becomes - : now
0
P=fJLfl' + \ -/(Z2 ^l - /li'2 COS ( - '),
but cos (0 - <£') is not greater than 1 ;
20
hence,
1 -/*/*'
or, squaring and reducing,
(fj. — p)* is not > 0,
which cannot be unless /u. = //, and this will give
cos (<p - <£') = 1 ;
and therefore 0 - 0' is zero, or some multiple of STT; hence
the series Q0 + 3 Qx + &c. is a discontinuous function, being
always zero, except when M = M' and <f> ~ <p' = 2wrr, (m
being any integer,) in which case it becomes -. It is evi-
dent, as in the former case, that this series is perfectly
discontinuous, being zero for all values of p that differ even
in the least degree from unity, and then when p = 1, sud-
denly assuming the form - .
25. Now, wherever we have occasion to use the series
Q0 + 3Ql + Sec., it will occur under integral signs relative to
fi and 0, and the limits of <f) will be 0 and STT; hence, by
what we have proved in the note* respecting the limits of
* If x be any arbitrary quantity occurring in any investigation, its differen-
tial dx may be defined to be any small increment of x, made use of with the
understanding that it is to be put equal to zero at the end of the investigation ;
and if f(x) be any function of at, its differential df(x) may be defined to
be the corresponding increment of /(#), that is,
The symbol j written before a differential, is generally taken to denote the
quantity from which the differential is derived, that is,
/«*/(*) =/(*),
and the notation /**<*/(*) is taken to denote the difference f(x3)-f(xl):
but this notation has a much more important signification : for in the equation
put for x successively the values .r,, Xi + dn, z1 + 2dn, &c., #, + (»- \)dx,
and add the results, and we find
! + dx) + d/(-r, + 2dx) &c. + df{Xl + (n - 1) dx\
21
integrals, 0 will receive all the values between 0 and 27r,
inclusive of the inferior limit, and exclusive of the superior,
and will therefore never actually be equal to, or exceed 2?r.
Also in all cases we shall be concerned with the same
may be supposed true of <p' ; for in all our investigations,
wherever 0' recurs, the value <f)' = 2 IT, or any greater quan-
tity, will be only a repetition of <f>' = 0, or some value
between 0 and 2 TT ; hence we may consider that <f> ~ <f>
never actually equals or exceeds 2-rr; and hence it will be
only for one value of 0 ~ 0', viz. 0, that p will become
unity ; hence, by what we have proved, the series
Q0 + 3Q, + &c.
in all cases we shall be concerned with, will be absolutely
zero for all values of fj. and 0, except the single values
[A. = p and 0 = 0'. We now proceed to prove some re-
markable properties of this series, which result from its
discontinuous nature.
26. From Art. 22 it appears immediately that
= 4<7T.
Now, here the quantity under the integral signs is,
as we have proved, always zero, except when p. = /m' and
0 = 0'; it is therefore no matter what the limits of the
integration be, provided they include between them the
or supposing >rj + ndx = ,r2,
df(xt) + df(xl + dx)+&c. till we come to df(xa-dx)=f(xs)-f(xl)
= £«<*/(*).
Hence it appears that if*8 df(x) denotes the sum of a series of values of df(x),
got by giving x all its values between the limits xt and xz inclusive of the
former limit and exclusive of the latter; that is to say, all the values of x
which form an arithmetic series whose common difference is dxy commencing
with xl and ending with xz- dx. The remark respecting the limits is im-
portant whenever discontinuous functions are concerned, as in our present in-
vestigation ; and we must remember that though the last value of x approaches
indefinitely near to x2, it never actually becomes equal to it.
alues (fji = /u') and (0 = 0'), respectively ; hence, if /*,, ^2,
i» 02*5 be any limits which do this, we have
* (Qn + 3Q! + &c. ...)dnd<b = ±Tr.
27- In the same manner, if F (n<p) be any function
of /u and 0, which is always finite between the limits - 1
and 1, 0 and Zir, F(yu</>)(Q0 + 3Q1 + &c.) will be always
zero, except when JM = // and 0 = 0', and we shall have,
as before,
/^2 rM
•^l »^l
28. Now let F(n"(f>") be the greatest value of F(^0),
between the limits /ui, yua» 0i> 02 '•> ano< let ^'(/*//0//) be the
least ; then it is evident from the nature of an integral,
considered as a sum, that
/-</>* rn* F ^/0/j ^QQ + 3 Q! + &c) d^d0
is not greater than
.F(/u"0") y T 2 (Q0 + 3Q, + ...) dyurf0,
and not less than
i. e. (by Art. 22),
not greater than 4 ?r F (^"0"),
and not less than 4 TT T'1 (^^0^) ;
and this is true, no matter how close together the limits
* Of course these limits are supposed to be included between - 1 and + 1,
0 and 2ir.
/"u Us* <£n ^a? be taken, provided /UL' and (j)' be included be-
tween them. Now /*", 0", and /u//5 0//s are also always
included between these limits ; hence, since /&'<£', M"> </>"»
yu//s 0/x, are respectively always included between limits
which we may take as close together as we please, it is
evident that we may suppose //', 0", and //„» 0y/» to differ
from yu'0' respectively by as small quantities as we please ;
and therefore, since F (/u. <p) is always finite, we may in
the above inequalities suppose F(/JL'> <^>") and F (p.^ <£„)
as nearly equal to F(fj!<p') as we please, which evidently
cannot be, unless
FT
jfr jfr
29. We shall give another demonstration of this
remarkable result.
Assume, as we evidently may,
(Q0+3Q1+ ...
Then, as before,
r* T
fc/Q •/ — 1
multiply this equation by d// d^)', and integrate between
the same limits, and we have*
It is necessary to take the same limits, otherwise in the integral
/i and ^>, the values of the variables for which Q0+3Qi + ... becomes «, will
not be always included between the limits, and therefore Art. 2fi will not apply
to it, and our proof will be incorrect.
24
Hence, by Art. 26',
differentiating this equation relatively to <p., and yu2 suc-
cessively*, we have
4-TT F (fJL2(f)o) =,/*(Al2</>2);
hence, since ^2 and 02 are arbitrary ,y= 47T-F; and therefore
(Q0 + 3Q,+ ...) dfjidQ
30. We may hence find the value of
TV' YnQndndd).
Jn J -i
Yn being any Laplace's coefficient ; for Yn being a rational
and integral function of /m, \/l-to>2, cos ^>, sin 0-f- will be
always finite ; hence we may put Yn for F (/a. (pi) in Art. 28,
and we find immediately by Art. 21,
PV YnQn
J0 J-i
* To shew how to differentiate a definite integral with respect to its limits,
let f(x) + C denote the indefinite integral off'(x), then
differentiating this relative to <r2, we have
and differentiating relative to
and in a similar manner we may differentiate integrals relative to two or more
variables.
t This is the reason why we have restricted Laplace's coefficients to be
rational and integral.
25
where Y ' denotes what Y becomes, when /tx.' and <f> are put
for (ix and (f> in it.
This is a very important result ; in fact, this, and that
in Art. 21, are the properties which render Laplace1 s
coefficients so very useful in integrations such as we have
to perform in Art. 15.
31. The equation deduced in Art. 28, interchanging
fjL and (f)' for /tx and <£, shews that if F(fj.(j)) be any func-
tion of fjL and (p, which is always finite, it may be expanded
in a series of Laplace's coefficients ; for by this equation
F (n<p) = a series whose general term is
2n + 1 rt tr /M , , ,
— / I F (/JL <p ) Qn d fJL U (t) .
4>ir J o »'— i
Now this quantity evidently satisfies any linear differ-
ential equation relative to fj. and 0 that Qn satisfies ; there-
fore it satisfies Laplace's equation of the wth order ; moreover
it is a rational and integral function of /x, \/l — ti2, cos <£, sin<^>,
for Qn is so, and
2W+ 1 /-8W /M r
will evidently differ from QB, considered as a function of
^, \/l-/x2, cos 0, sin 0, only in having different coefficients
to the powers of these quantities ; that is to say, if A'
be any coefficient in Qn, then the corresponding coefficient in
') Q. dp' dip
27T
will be
hence the several terms of the series to which F'(/u'(p') is
equivalent are rational and integral functions of /u, \/l-/r,
cos 0, and sin 0, which satisfy Laplace's equation, and
4
are therefore, according to our definition, Laplace's co-
efficients.
32. No function can be expanded in more than one
series of Laplace's coefficients.
For, if possible, let
Y0 + Yl + ¥„ + ... + &c. and Z0 + Zl + Z2 + &c.
be two different series of Laplace's coefficients equivalent to
the same function, then, since this is the case, we have
Y0 - Z0 + Y, - Zl + Y, - Z2 + &c. = 0;
multiplying this equation by Qnd/i.d<p, and integrating be-
tween the limits - 1, 1 ; 0, 27r, we find (by Art. 22 and 30),
(Y'n -Z'a) =0,
"2n + 1
hence Yn = Zn ; and therefore the series are the same, and
the function can be expanded only in one series of Laplace's
coefficients.
33. The conclusion we have just arrived at seems to
be at variance with the fact that the quantity
may be expanded in two distinct series of Laplace's co-
efficients, viz.
but this is only an apparent discrepancy, for
the function to be developed, admits of two values, one
positive and the other negative, on account of the ambiguity
of the sign of the square root ; and therefore ought to admit
of two developments. That the above are the two develop-
ments corresponding to the two values of the square root,
will follow from the following proposition : viz.
27
34. To determine the sign of the series
This series being the development of the expression
1
it is evident that it can never become 0 as long as h is not
infinite, but it will become infinite when
1 - 2ph + h2 = 0 ;
2
that is, when p
.
Now 1 + h2 is always greater than 2A, except when h « 1,
in which case 1 + A2 = 2A; hence, since p, being a cosine,
can never exceed unity, this equation can only be satisfied
by h = 1 , and therefore p = 1 ; hence
cannot become infinity unless h be unity ; hence, if h be not
infinity or unity, this series can never become zero nor in-
finity, and therefore can never change its sign : supposing
then that h is not infinity or unity,
will always have the same sign whatever be the value of p.
Hence, supposing p = 1, and therefore Q0 = 1, Q1 = 1, &c.
(Art. 16.) it is evident that
Qo + Qih + &c. has the same sign as 1 + h + h? ...
1
i. e. as - -;
1 -h
hence, if h be less than unity, this series is positive, and if
h be greater than unity, it is negative.
In the same way it may be proved that
tilc samc 8'n as
\ 7 g -r* 7 - '
rtf ft ft -~ 1
and is therefore always negative when h is less than unity,
and positive when h is greater than unity.
Hence, when h is less than unity,
i
QO + Q\h + Qzh2 + is the positive value of
V 1 - 2ph + tf
and - \ Qo + Qi 7 + . . . \ the negative value ;
h ( h }
and when h is greater than unity, the reverse is the case.
35. To represent, then, the true general development
of —j- , let A; be a discontinuous function of h,
Vl — %ph + h*
such that A: = 1, when h is less than unity;
and k = 0, when h is greater than unity.
Then, whatever h be, the positive value of — ^
V ]
will evidently be
and the negative value will be
(1 -k} {Q0+ Q,h + Q2A2+ ...} + k JQ0^ + Q ^+ ...1.
Thus, in reality, , , if we restrict our-
Vl-2ph + hz
selves to only one of its values, can be developed in only
one series of Laplace's coefficients. The conclusion we have
just arrived at respecting the true development of this
quantity, will be highly important in our future investi-
gations.
36. We shall conclude this chapter with the following
important property of Laplace's coefficients, viz.
29
If Y0+Yl + F2 + &c. = 0 (l),
be any equation arranged in a series of Laplace's coeffi-
cients, IJL and 0 being indeterminate, then we must have
F0 =0, Y,= 0, F2 = 0, &c.
for, multiplying (l) by Q,,dfj.d(p, and integrating between
the limits - 1, 1, and 0, 27r, we have, (by Arts. 22 and 30,)
2tt + 1
and therefore Yn = 0.
We have now considered these remarkable functions at
sufficient length for our present purpose ; we shall recur to
this subject in Part u. of these Tracts.
CHAPTER III.
FIGURE OF THE EARTH.
37. WE are now prepared to return to the equation
(5'"), see Art. 15 ; the properties we have proved La-
place's coefficients to possess, will enable us to perform the
integrations relative to // and <£' with great facility.
In Art. 16 we stated that we should expand the quan-
tity f (a, a'), or / , in either of the series
Now there is an ambiguity in this quantity, since on account
of the square root, it admits of two values, one positive
and the other negative ; but, on referring to Art. 6, it is
evident that the square root is always supposed to have its
positive value; for the distance between $m and $TW', which
is expressed by this square root, is evidently taken to be the
absolute or numerical distance, without reference to sign ;
hence, by what has been proved in Art. 35, neither of the
above series will give the true general development of
y(«, a'), but we must put
/(a, a') = *. Q0 + Q + Q2 + &c.
+ (1 - fc)a' JQ0 + Q,^ + Q2^+ &c.},
where k is a discontinuous function of — , which is always
unity while a is less than a, and zero when a is greater
than a. For brevity we shall take An to denote the co-
efficient of Qn in this series ; i. e.
and then, we shall have
38. What we have proved of Laplace's coefficients
naturally suggests the advantage of expanding the quan-
tity u in a series of Laplace's coefficients ; this may be
done, since, on account of the assumption we have made
respecting the nearly spherical form of the strata, u can
never become infinite ; we shall assume, therefore,
u = u'0 + u\ + u'2 + &c.
tt'o, M'J, &c. being Laplace's coefficients of the orders 0,
1, 2, SEC. and functions of a', /u', and (£'.
We shall denote by uot ult u2, &c. what w'Q, u ^
u'2, &c. ... become when a, yu, and (f> are substituted for
o', /n', 0', and therefore we shall have
u = u0 + ul + u2 + &c.
We shall also have occasion to make use of the values
of w'0, 7/n w'j>, Sec. when /u. and (f) alone are put for /u.'
and 0', a' remaining unaltered, These values we shall
denote by 'w0, fuiy 'w2» &c-
39.
find
and
J9. Hence in the equation (5'"), see Art. 15, we
immediately by Arts. 22 and 30,
f
32
and hence, putting u0 + w, + w2, &c. for u,
y _ 47r / -• y i ^ . 2 _ /.. ... . .. . o— \ a^o
= 4?r / ' p A0da
•/«
+ a series whose general term is
4-7T <y* f ' p' { aun — — + - — -. (a An'un)\da .
Jo \ n da 2n + 1 do' v J
40. If we put for A0 and .4n their values, viz.
a'2 an+2 a"
k — + (l-k)a', and k — + (1 - A?) ^ ,
(see Art. 37.) ; the factor of 47rfc>2 in this general term be-
comes
Now /c is always unity while «' varies between the
limits 0 and a, and zero while a varies between the limits
a and al (a being of course never greater than a^ ; hence
that part of the quantity under the integral sign which is
multiplied by k will not exist except between the limits
0 and a, and that part multiplied by 1 - k will not exist
except between the limits a and al ; hence this integral may
evidently be put in the form
-UJLf* p'a'2da'+-r ~ r (a p — (XXn+s) da
a Jo ^ (2n + 1 an+l J0 r da' y
t , d f 'u. \
p —, \~-. }
^ da \an-2J
33
We shall, for brevity, denote this expression by
<*n "„,
<rn being merely a prefix assumed to express a certain
operation performed on WB.
In the same way the first term of F, viz.
4-Tr Ip A0da',
"0
will evidently become
4>7r Ty { k — + (l - k) a] da,
or
4>Tr J p a'2 da + 4nr f l p a da .
Hence the development of V becomes finally
F = — fa pa2da' + 4>7r / ' pa da
a Jo r Ja (
4- 4-TTca2 cr0w0+ ff\^\ + cr2w2+ Sue.
It is evident that this is a series of Laplace's coefficients,
the sum of the first three terms being a coefficient of the
order 0, and the succeeding terms of the order 1, 2, 3, Sec.
respectively.
41. We shall now express the value of Fgot from the
equation (A") (see Art. 13) in Laplace's coefficients; to do
this we observe, by trial, that -^ — /m2 is a Laplace's co-
efficient of the order 2 ; hence the value of V got from (A")
will be arranged in Laplace's coefficients by simply putting
it in the form
34
42. If we now subtract the two equations we have
thus obtained, we shall eliminate V and arrive at an equa-
tion consisting of a series of Laplace's coefficients, which,
by (Art. 36), must be put separately equal to zero; hence
we get the following equations, viz.
— 'o'^d' '
l pada' + 4>Tra)2(r0u0 — C + - = 0,
a Jo ' Ja ' 2
— (^ - fjf) = 0,
and a-nun = 0, for all values of n except 0 and 2.
Thus we have eliminated V without knowing what
function it is, and obtained equations for determining w0,
MJ, uZ9 &c., and therefore the equation to any stratum.
43. We shall now proceed to solve these equations,
commencing with the last of them, viz.
<rnun = O...(l), except n=0 or 2.
By (Art. 40) this equation is equivalent to
-^ rapfartda' + - - l— - - fa p-^.(a'n+*fun)da
a Jo ^ (2n + l)o"+l./o ^ da
a" ro, , d ( '«. \ ,
/ P -r-7 -r-i )da = °-
1 Jo r da k«*~y
+
2n +
Now, by Note, p. 24, if we multiply this equation by aw+1,
and differentiate it relatively to a, and then multiply it
by — , and differentiate it again relatively to a, we shall
by this process get rid of the two last integral signs, and
arrive at a differential equation which may be put in the
form
• dun
A +4...0 ...... (2),
35
Alt Ay being functions of a and p, which we have no oc-
casion to determine. Now let v and v' be two quantities
which satisfy the equations
crnv — an =
1 ......... (3),
and let C and C" be two constants, then
un = Cv + CV
will be a solution of the differential equation (2) ; for, per-
forming the operation an on both sides of the equation,
un = Cv + CV,
we have &nun = crn(Cv + CV)
= C(TnV + C'(Tnv'
(evidently from the nature of the operation <rn)
or ffnun = Ca" + C' —^ , by (3).
Now if we get rid of the integral signs in this equation by
the same process we have applied to the equation (1), the
second side will evidently be made zero by the differenti-
ations, and thus we shall arrive at the same differential
equation as before; hence
»,-Ca + CV«
is a solution of (2). Now this is evidently true whatever
be the values of C and C"; hence this solution contains
two arbitrary constants, and is therefore the most general
solution that (2) admits of. Hence all values of un which
satisfy (l), since they also satisfy (2), must be values of
* It is evident that v and v' are two different functions, otherwise the
equations (3) would give
which is not the case. This remark is necessary, for if v and t>' were not
different functions then C and C' would add together, and therefore be equi-
valent to only one constant.
Cv + C'v. To determine what values of Cv+C'v' satisfy
(1), substitute Cv + C'v for un in (1), and we find
or Ca" + C' — = 0, by (3).
Now this equation ought to be true for all values of a ;
hence (7 = 0, and C'=0; hence it is evident that only
one value of Cv + C'v, namely zero, satisfies (l); and
hence it follows from the equation (l), that
Thus un is zero for all values of n except 0 and 2 ; this
result produces a considerable simplification in the equation
to the strata.
44. We shall next consider the equation involving w2,
which may be written thus,
In this equation we may conceive w2, which is a function
of n, vi - ^, cos 0, and sin 0, to be developed in a series
of powers of i — /j? and <£. Let y (^ — yuc)m0" be any term
in this development, -y being an unknown coefficient to be
determined, then the corresponding term in the equation
will be
except m = 1 and n = 0, in which case it will be
Now i — /a? and 0 are arbitrary ; hence we must put the
coefficients of their several powers separately equal to zero ;
and hence for all terms, except that in which m = 1 and
n = 0, we have
0"27 = ° i and therefore y = 0, as before in (Art. 43) ;
37
and for the term in which m = 1 and n = 0,
a2
ff*V + ^.= \
which equation will determine y ; hence we have
w2 = 7 (i ~ M8)»
where -y is given by the equation
45. We cannot determine UQ from the remaining equa-
tion in Art. 42, on account of the. unknown constant C in-
volved in it ; but the value of u0 follows from the assump-
tion that we have made respecting a in (Art. 12) ; namely,
that it equals the radius of the sphere of the same capacity
as the stratum whose parameter it is: for this assumption
gives
— => volume included by stratum,
3
rtir r 1 /«
I. Li
a3 rz-T
jjf
a(l+a>8u)
47ra3a>8w0> by (Art. 22.)
Hence UQ = 0.
46. Thus we have determined the values of w0, M,,
w2, &c. ; and it now only remains to substitute these values
in the equation,
r = a . { I -i- or (MO + ut + u2 &c.) | ;
38
and we find that the equation to any stratum whose para-
meter is a, is
Now the equation to a spheroid generated by an ellipse
revolving about its minor axis, is
V 1 - e2cosa l^ - 9\
a being the major axis, e the eccentricity, and 0 ,the angle
which r makes with the minor axis. Supposing e very
small, this equation becomes
e2 e~
r = a \l + ~sin20*
2 2
= a {l -€cos20j,
e being the ellipticity,
maior axis — minor axis
for the ellipticity = — ? : :
major axis
-, nearly.
This equation may evidently be made to coincide with
the equation to the stratum, by putting
a e = o to2 ;
to2'/
and therefore e = - :: — = ft)2/Y» nearly.
Hence the strata are all spheroids of revolution about
the polar axis ; the ellipticity of any stratum being ufy,
y being got from the equation
o2 _
STT
or putting in this equation — for y, the equation for deter-
(t)
mining the ellipticity of any stratum, will be
8?r
or, by Art. 40,
— / p a da' H / p' — "' (a'5e')dar
a Jo 5a Jo da
a8 /-"i ,de 0*0?
+ — p — ; da + = 0,
5 Ja ^ da' STT
which equation will give e when p' is known as a function
of a'.
47. Thus we have arrived at the remarkable result
that the mass must be arranged in strata of equal density,
which are all spheroids of revolution about the axis of rota-
tion, their ellipticities being connected by the equation just
obtained.
It is evident that our investigation gives us no infor-
mation respecting p ; hence the law of density of the strata
is quite arbitrary, and must be determined, if possible, by
some independent method.
48. We shall presently shew that the results of our
investigation may be compared with observation in a most
satisfactory manner, without knowing any thing of the law
of density of the strata. This is fortunate, as we have no
means of determining this law, but must have recourse to an
hypothesis which it must be confessed is rather empirical ;
but as the results it leads to may be made to agree well with
observation, we must look on it as probable. The hypo-
thesis we allude to is this, that the variations of the pressure
in the interior of the earth (supposed fluid and of the same
chemical constitution all through,) are proportional, not to
the variations of the density as in gases of uniform tempe-
rature, but to the variations of the square of the density ;
i.e. that instead of having dp = kdp, we have dp = kpdp.
There is some slight reason a priori for assuming this
formula, for it is evident that p ought to increase more
rapidly with p in the fluid composing the earth, than it
would do in gases, both on account of the incompressibility
of that fluid, and the increase of temperature as we go
towards the centre ; and hence kp dp will represent the
variations of p better than kdp. But the chief reason for
assuming this formula is, that it leads to correct results, and
simplifies the equations we shall be concerned with, as will
appear.
49- Assuming then this connection between the pres-
sure and density, we may calculate the law of density from
the equations
.................................... (1),
V = ^-J^ p'a'da'+tTT rip'a'da ......... (2);
which are got from (Art. 6) and (Art. 42) ; neglecting or,
for the centrifugal force will make a very little difference
in the law of density, and it will be useless to be very
accurate here, as we are proceeding on rather uncertain
grounds.
We have, from (l),
dp_ dV
d^** P da
multiplying this by — , and differentiating relatively to a,
P
we have
d
Substituting the assumed value of dp, viz. dp — kpdp,
we have
which may be put in the form
Therefore, putting — = g8,
pa = A sin (70 + 5),
sin
and p =
A and B being arbitrary constants.
50. To determine B, let a = 0, then we have
p = co , unless J9 = 0 ;
hence since p, as we may evidently assume, is not infinite
at the centre, B = 0, and we have
A sin qa
P ~ a
To determine A and q, let pj be the density of the superficial
parts of the earth, i. e. the value of p when a — a, ,
and let D be the mean density of the earth ; then
mass of earth
its volume
6
47T
42
/ pa* da'
JQ
s
3 A
3A ra, .
= — — I a sin q a da ;
a3 J0
"T ^ sin go'
3A
- .
also 0i = — sin
a,
hence
sin 9^! — qal cos graj 9 a\ JJ
from which equation q may be found, and then we shall
have A from the equation
sin 9 a,
51. Observation shews that — is about — , and on
D 11
substituting this value of --j- in the equation for deter-
mining (7, we shall find by repeated trials (which is the
only way we can solve it) that it admits of several solu-
tions, of which one only leads to right results; it is this,
<7«i = about — ;
1 5-7T .
hence, substituting — — for (7, we have
. . /57T a
A sin | — —
o a.
43
To determine A, we shall put a = a, in this, which gives
A . 5-n-
« sin T
- sin 30°
and hence .4 = S^e^, and we have
O
which, if our hypotheses be correct, expresses the density
of any stratum in terms of the parameter of that stratum,
and the superficial density.
52. The method by which we have arrived at this
formula for the density is not very satisfactory, and we
shall therefore consider it as empirical ; we observe that it
gives a density which increases as we go towards the centre,
but does not become infinite there; this is most probably
the case; it also makes the pressure vary more rapidly as
we approach the centre than it would do if the earth were
gaseous and of uniform temperature; this is also most
probably the case; and it gives the mean density of the
earth its proper value: we shall prove presently that it
gives the true value of the earth's ellipticity, and also the
true value of the coefficient of precession; hence, on the
whole, we may assume it with some probability as the law
of density.
53. Hence, finally, it follows from the hypothesis of
the earths original fluidity ;
(1) That the earth ought to consist of equidense
strata, all spheroids of revolution about the axis of
rotation.
44
(2) That if e be the ellipticity of any of these strata
it satisfies the equation
- i/v«w + A/V/-- <•• vv°'
a Jo r 5a?Jo r da
a2 /">i ,de . , a2
'
a /">i ,e . , a <t)
+ — / p — , da'+ - = 0.
5 > r da STT
(3) That we may assume with probability the law of
density to be
a /STT a\
p = 2/0, - sin — - ) .
ri at \ 6 aj
CHAPTER IV.
METHODS OF COMPARING THE RESULTS JUST ARRIVED
AT WITH OBSERVATION.
54. IN order to test the correctness of the conclusions
we have just arrived at, we shall now deduce from them
results of a more practical character, which shall admit of
direct comparison with observation : the first result we shall
deduce is this ;
If * be the length of a meridian arc, measured from the
pole to any place whose colatitude is c, then
a and ey being the major axis and ellipticity of the earth.
For, by Art. 48, the equation to any meridian is
r = a (1 - e, cos2 0)
= «(l--cos20) ......... (1).
XT ^^
Now_= r*
= r, neglecting squares &c. of e,
_«{!-!.<_ | COg 20} by (,),
therefore, integrating,
(2).
46
Adding no constant because « evidently = 0 when
0 = 0.
Now c (the colatitude) is the angle made by the normal
with the polar axis ; hence (9 — c) is the angle made by the
normal and radius vector, and hence
or, neglecting the squares &c. of small quantities,
9-c = e sin 20, by (1),
therefore
9 = c + e, sin 2 9
= c + et sin 2 (c + e sin 20)
= c + e/ sin 2c nearly ;
hence, substituting in (2), we have
a \c + e sin 2c -- - c — - sin 2c|,
2 4
55. To shew how this result may be compared with
observation, let s' and c' be the values of s and c correspond-
ing to another place near the former, and on the same
meridian, then
and therefore
*' - s = a
j 1 1 - |) (c' - c) + ^ (sin 2c' - sin 2c)| ,
cos ''+ c> sin c'-6''
47
Now in this equation s' — s, being the distance between
two places near each other, may be determined by the usual
method of triangulation ; cos (c + c) may be found by any
of the ordinary methods of determining the latitude of
places, without aiming at any great accuracy, since it is
multiplied by the small quantity e ; c' — c, not being mul-
tiplied by a small quantity, must be determined more ac-
curately by observing the meridian zenith distances of the
same star at the two places, and taking the difference
which will evidently be equal to c — c ; thus we may put
our equation in the form
when A, B, C are known ' quantities got by observation.
In the same way, by observations at other places, we
may obtain any number of similar equations; suppose
them to be
&c ....... &c .......
From any two of these equations we may determine
a and e, ; and the values of a and e so determined* ought to
satisfy all the other equations; hence, if we find that all
these equations are satisfied by the same values of a and ey,
it is evident that our result agrees with observation.
56. Now a number of meridian arcs have been mea-
sured, and a system of equations similar to the above have
been formed, and it is found that the values
a = 3.Q62.82 miles, e = — ,
' 306
satisfy them all to a remarkable degree of accuracy, allowing
for certain small errors which may be easily accounted for ;
and which, even considering them in the most unfavourable
point of view, are very much smaller than e,, which is itself
* Or rather, the values of a and e, got from all the equations by the method
of least squares.
48
a very small quantity ; and indeed if we bear in mind the
delicacy and number of the observations requisite in order
to form the above equations, the smallness of the errors
is most remarkable.
57. On the whole we are justified in concluding from
observation, that the equation to any meridian and there-
fore to the earth's surface is very nearly this, viz.
r = a { 1 -e/cos80},
where a = 3902.82 miles,
and € — — .
' 306
Hence the hypothesis of the earth's original fluidity
leads to a very peculiar result, which is capable of varied
and extensive comparison with observation, and which
agrees with it in a remarkable manner; from which we
must conclude that this hypothesis is most probably true.
58. The second result we shall deduce from our
theory is this ;
If g be the force of gravity at any place whose colati-
tude is c, then
g = U [ 1 + I — 61 I COS'C^ .
Where G is a constant, namely, the value of g at the
equator, and m the ratio of the centrifugal force to gravity
at the equator*.
* To determine w, we observe that
G - 32.2 + a small quantity,
hence
ura
32.2 + a small quantity
In putting 32.2 for G we have assumed a foot to be the unit of length,
and a second the unit of time; hence we must express a and to2 in terms of
these units, and therefore we have
a = 3962 miles, nearly,
= 3962 x 5280 feet,
and
49
To prove this, we observe that -g is the resultant
of the forces which act on a particle at the surface, which
forces are, by (6),
dV dV dV
— + ftT<r, — - + ory, — — ,
das dy dz
and therefore g must balance these forces ; hence, by the
principle of virtual velocities, if dr be any variation of
r, and da?, dy, d#, its resolved parts along the axes of
co-ordinates, and >// the angle which r makes with the
direction of g, we shall have
.r ) dx + ( -- \- u?y }dy + — dsr,
\dy / * dx
which gives
I
\d.T.
"
r
Now g acts in the normal (by the principles of
Hydrostatics), and the normal evidently makes a very
small angle with the radius vector ; hence \|/- is very small,
V
and therefore, since cos \J/ = 1 - — + &c., we may, neglect-
ing squares and higher powers of small quantities, put
cos \js = 1 ; moreover we have, by (6) and (42),
and
4-Tr r<i , ,n , , /*", , . , , w2a2
C = — / pa* da + 4-TT p a da + - ;
a Jn f Ja ' 3
Htld "> = 24ho^
2-r
24x60x60'
Mnkins; the<:e substitution*, we shall find
m = about — — *
50
hence, observing that a is a function of r in virtue of the
equation
r = «{l+e(i-M2)},
we have, substituting in (1),
dC
g=-dr
_dCda
dr dr
*TT ra , , 2ft,2 a\J d(ae) \
a da — l ~ - (-M
which, neglecting the product of small quantities, may
evidently be put in the form
where
G ** -— I p'a'ada' + small quantities.
Since g *= G when /u = 0, it is evident that G is the
force of gravity at the equator.
(Of course, in all our formulae, at is supposed to be
put for a after all differentiations and integrations have
been performed, since the particle on which g is the force
is supposed to be at the surface.)
Now by (53),
-1 (ap'a'zda'+~ ^ o'-^
a Jo P 5a5 Jo r da
a2 /*«i ,de , ,
+ — / P -r-, ; da + - = 0.
5 Ja ? da STT
Multiplying this equation by a3, and differentiating relative-
ly to a, we have
d(ea?) ra , , ra, ,de , 5ft,2 a4
- / oVW+a4 / o— ,da'+- - = 0,
da Jo r J" r da STT
51
or, observing that af is to be put for a,
8?r / p a> da
_ 5g,V
= TcT'
since G = ~ f" pa'*da + small quantities,
fl ''O
5ma
= 5
centrifugal force a>2o
since m = ; = — - ;
gravity at equator G
hence — = — - e
da a da
5m
and hence equation (l) becomes
since 0 = c + esin2c, by (64).
59- To shew how to test this result by observation,
we observe that if p be the length of the seconds pendulum,
then since 1 = 27T \/- , we have p = ^L
x-i
when P = — 2 = value of j» at the equator.
52
Now p may be determined by observation at any place ;
hence, by observations at different places, we may find a
system of equations such as before ; viz.
See. &c.
when Ay B, A', B"> SEC.... are quantities got from observation.
Now it is found that the values
, 5m
P = 39.01228 inches, and -- e/ = .005321,
satisfy all these equations, not so exactly as before, but yet
with remarkable accuracy, considering the small quantities
we are engaged with; hence this result is another proof
of the Earth's original fluidity.
60. The comparison of this result with the former is
a strong additional proof; for the former result gives
and since we know that m = - , the present result gives
this is a very remarkable coincidence, and must be consider-
ed as a decided proof of the correctness of our hypothesis.
61 . If G' be the value of g at the pole,
G'=G
_
and
53
G'- G 5m
/
This result is Clairaut's theorem.
62. We may determine the Earth's ellipticity by means
of the law of density assumed in Chap. in.
The equation for determining e is
- 1 f'p'aPda' + ~ f V J~, (ea*)daf
aJ0 r 5asJ0 r da
a /•«, ,de' , «r a*
+ -/ P^-7da'+ -- = 0.
5 Ju r da 8-rr
Integrating the second and third terms by parts, we
have
a2 r^du , , ,
—
a r^u , , , of a
-- / — , e da + L- -
5 Ja da 5
5 Ja da 5 8?r
A sin qa
Now putting p = — — , we have
/ p'a'~da'= — (sin qa — qa cos qa)
dp A ,
-1-7 = -- - (sin q a - qa cos qa ) ;
hence, if we assume
e (sin qa - qa cos qa) = »;,
(1) will become
At) A ra , t , A a* /-a, 1J. . , , O€. (O*\ ,
--5-+— ,/ via*dc!+— - -±da+(^+ — a2 = 0;
(f- a 5aiJo 5 Ja a'* \ 5 S-rrJ
dividing this by a2, and differentiating relatively to a, we
have
A d / ri\ A ra , ,
- - — [-J - - / na da '= 0 ;
tfda\\arj erJ*
54
or multiplying by a6, and differentiating again,
63. To solve this differential equation we shall assume
£' such a function of a that
therefore
Hence, substituting in the equation (2), we have
a??-3a fVr'da' + (fa /V Fa'? da'* ;
»/0 ''O •'O
or dividing by a and differentiating,
*g-«i;+ *".f *?'«•:
or again dividing by a and differentiating,
hence <£= C sin (qa + C'), C and C' being arbitrary
constants, and
r a'^da = — {sin (qa + C') -qacos(qa + C')},
and
{"a faa<gda' = — {3 [sin (qa + C') - qa cos (qa + C')]
55
Hence we shall have, putting C instead of — ,
rj = C | -y-j [sin (qa + C')-qa cos (qa + C")] - sin (qa + C')> .
64. We might determine the constants C and C' by
substituting this value of rj in (l), Art. 62, but the following
method will be more simple : in the first place we may see,
a priori, that C' must = 0, for otherwise we should have
»/, and therefore e very large when a is very small, contrary
to our assumption of the nearly spherical form of the strata ;
hence
rj «• CJ-jj (sinqa - qacosqa)- sin qa > ...... (2),
and therefore, since rj = e (sin qa — qacosqaj,
. --
q2a2 sin qa - qa cos qa)
i s 2\
We shall determine C by means of the value of ^ea ' ,
da
got in p. 51.
Multiplying (2) by a2, and differentiating relatively to
a, we have
d(r,a2)
— = Ca (sin qa — qa cos qa),
da
also doing the same to the equation
tj = e (sin qa— 70, cos qa),
we have
— ; = — i (sin qa - qa cos qa) -f eg2 a3 sin qa.
da da
Equating these values of , we have
da
1 d (ea8) sin qa
'+^0*^
oa sin ga - qa cos 70
56
. d(ea3) 5 ma
Hence putting a = a , and --- - = - •* , see p. 51,
5m 36/p,
= TH -~D~'
also putting a = nf in (3), we have by Art. 50,
from which two equations we get
5m
"i"
»H- D
XT 1 5ir p 5
rsow m = , qa = — , — = — ,
289 ' 67) 11
substituting these values and reducing, we find
e = about — .
306
This result agrees with observation, but the agreement
is not of much value on account of the assumption of the law
of density.
65. The effect which the attraction of the Earth has
on the Moon's motion, is usually brought forward as an-
other means whereby we may test the correctness of the
hypothesis of original fluidity ; and the agreement between
theory and observation in this particular is considered to
afford additional evidence of the truth of the hypothesis.
We shall attempt however to prove that this is not the case.
To do this, we shall shew that if the equation to the Earth's
surface be known, and also the law of variation of the force
of gravity, then the effect of the Earth's attraction on the
Moon follows as a necessary consequence, independently
of any theory except that of universal gravitation.
57
66. It is evident from the smallness of the variations
of the force of gravity, that the Earth must consist of
nearly spherical strata; hence all the results we have al-
ready obtained, so far as they depend on the nearly spheri-
cal form of the strata, will be true whether the hypothesis
of original fluidity be correct or not.
Hence, as before in p. 49, we shall have
putting a/ for r in the small term.
* Now the expression for V in Art. 40. may evidently be
put in the form
a series whose general term is
+ 47TO)2
, d
hence, differentiating relatively to r, and observing that a is
a function of r in virtue of the equation r = a (l + o>2w), and
supposing that at is put for a after the differentiations, we
have
flTF__47r /.Ol , , /47r „ \ da
~a7~~^J0 pa
* For the part of V which is multiplied by 4ir £p'a'eda' is evidently
,&c.)}, which =-.
58
a series whose general term is
( a series wnose general term is \
1 (da
<. n + 1 ra, d „ , , , , > ,
\--r - -= / P — ,(una*+3)da + pauA dr
I 2n+l«n+2 Jo r da'v ' ' ' ")
,
+ 47ro>z
-
(2n+l)«<
Now since r = a (l + ct>2w),
c?a
and therefore — =1 - &>
da
we have, evidently, neglecting o>4 &c.,
MTT \ da _ f
\rt ?' ' * ') dr P' '
and therefore, neglecting a>4 &c.,
4£--?r>w
a series whose general terra is
47TO)2
hence if, for brevity, we put
shall have
-o>2ai (1 -/x2);
and hence
47Tft)2 , 2 3
— 5- { za + - z, + -5 z2 + &C. 5
59
Now if we suppose rj and g known, it is evident that
the first member of this equation will be known, and may
therefore be supposed to be expanded in a series of known
Laplace's coefficients ; and hence, since the Laplace's co-
efficients of different orders on each side of the equation
must be separately equal, by Art. 36, the values of Z0 Zl Z2
&c. will be known.
* Now it is evident immediately, from Arts. 14. and 40,
that the value of V for any external point is
V= — faip'a'*da + 47rte)2(- Z0 + - - Z, + 4 Z2 + &c
r Jo [T r r
hence, since Z0 Zt Z2 SEC. are known, as we have just
proved, it is evident that the value of V for any external
point is also known.
67. Hence, if we know the form of the Earth's sur-
face, and the law of the variation of gravity, we shall
know the value of V for any external point, and there-
fore be able to determine the attractions of the Earth on
that point without making use of the hypothesis of original
fluidity.
Hence it follows that if the form of the Earth's surface
and the law of variation of the force of gravity, calculated
on the hypothesis of original fluidity, agree with observa-
tion, then the effect of the Earth's attraction on the motion
of any external body, such as the Moon, calculated on the
same hypothesis, must also agree with observation, whether
that hypothesis be true or not; and hence we conclude,
that the motion of the Moon does not afford additional
evidence of the Earth's original fluidity.
* For the only difference made in the reasoning in Art. 40, by using the
expression for V given in Art. 14 instead of that given in Art. 15, will be
simply this, that we shall have to consider — instead of — ; also, since the attract-
ed point is external, and therefore r always greater than a', it is evident that k
will be always unity.
In the next chapter we shall determine the equations
of motion of a rigid body round its centre of gravity, and
thence deduce the Earth's motion round its centre of gravity.
We shall find that the result affords a confirmation of the
law of density assumed in Chapter in., and also of the
hypothesis of original fluidity.
CHAPTER V.
EQUATIONS OF MOTION OF A RIGID BODY ROUND ITS
CENTRE OF GRAVITY.
68. WE know from the principles of Dynamics, that
a rigid body acted on by any forces moves relatively to
its centre of gravity, in the same manner as if that point
became fixed, all other dynamical circumstances remaining
unaltered; hence, whenever we wish to investigate the
motion of a body relatively to its centre of gravity, we
may consider that point as fixed, and this will render the
investigation simpler.
Suppose, then, that we have a body whose centre of
gravity is fixed, acted on by any forces ; let §m be any ele-
ment of it, any % the co-ordinates of Sm at the time t referred
to any arbitrary rectangular axes fixed in space, and origi-
nating in the centre of gravity ; let L M N be the moments
of the impressed forces round the axes of a? y % respectively,
then we have, by the principles of Dynamics,
tfco
— - * - -— « U M
(A).
69. In order to perform the integrations denoted by
2 in these equations, we shall introduce new variables
instead of xyz, which shall have reference merely to the-
position and motion of the whole body, and not to any
particular particle of it. To do this,
Let x y' % be the co-ordinates of $m referred to any
arbitrary rectangular axes fixed in the body, then we have
x = se cos (x x) + y cos (y' x) + % cos (% x) ;
differentiating this relatively to #, and observing that x y %'
do not vary with #, we have
dot , d cos (a?' a?) ,d cos (y' x) , d cos (z'x)
dt=* — dT-~ + y~— dT~ +%~~dt — '
Now the axes of x' y % are perfectly arbitrary ; we
may therefore suppose them so chosen that they shall coin-
cide with the axes of x y % at any instant we please.
Suppose therefore that this coincidence takes place at the
time #, then we shall have x' = x, y = y, % = % ; and if,.
„ d cos (x'x) d cos (y'x)
for brevity, we denote the values of -^ — - , - y— - ,.
at at
~ — -, at the instant of coincidence by X X' X" re-
dt
spectively, our equation becomes
dx
— = \x + \y + \ #;
at
X X' X" are evidently variables which have reference to the
position and motion of the whole body, and not to any
particular particle of it, for they depend simply on the
angles which the two systems of co-ordinate axes make
with each other at any time, or rather upon the rate at
which these angles are varying at the instant of coincidence
of the axes.
In the same way we may prove that
dy
-j~t = M + ft % + fji 'x,
dss „
— - = vss + i >x 4 v y;
at
where /M // n", v v v" are quantities similar to X X' X".
63
Now since $m is rigidly connected with the origin, we
have
xz + if + s? = constant,
dx dy dx
and therefore x— +y— + ss— =0;
dt dt dt
„ dx dy dz
substituting in this equation the values of — -~- —
dt dt dt
just found, we have
Xa?2 + fiy2 + v%* + (X'-f fj.")xy+ (p +v")yz + (v + X'')#a? = 0;
hence, since xyz are arbitrary, we have
X' + JM" = 0 // + v" = 0 v' + \" = 0.
. dx dy dss ,
Hence the values of -p- — — become
dt dt dt
-
dy
dx
— =vy-Xx.
To conform to a common and convenient notation, we
shall put o>j ft>2 0)3 instead v" X" /^L" respectively, and write
these equations thus,
dx
— .„,»-«*
— - = W3ar-tt,2f
at
we shall presently determine what ojj w2 o>3 are.
These equations express the relations which exist be-
tween the velocities of any element of the body and its co-
64
ordinates at the time £, in consequence of the rigidity of the
body. The substitution of these values of — — —
at at at
in the equations (A) will be very advantageous, since
o>i c«2 ft>3 are independent of x y z, and may therefore be
brought outside the integral sign 2. To perform this
substitution we have from the equations (J5), differentiating
dx dy dz .
and m the result putting for — — — their values given
dt dt dt
by the equations (J5),
d<o3
Now the axes of so y % are perfectly arbitrary, we may there-
fore choose them so that the principal axes of the body
shall coincide with them at the time £, and therefore we
shall have at the time #,
and hence, from the equations just obtained, we have at
the time #
at
dt dt
and hence the first of the equations (A) will become
~^m (a? + f) + w^ 2$m (* - y*) = N.
Now the principal axes coincide with the axes a? y z at
the time t, hence if A B C be the principal moments of
inertia of the body, we shall have
65
23m (x* - y2) =
(y* + *)
and hence ^. — — \- (B — A)
AT.
We may transform the other two of the equations (A)
in exactly the same manner, hence, instead of the equations
(-4), we have the following three equations, in which the
integral sign 2 is got rid of, viz.
A — + (C - B) ft>2o>3 = L
at
at
70. It remains to determine what wl w2 w3 are; the
equations (B) will enable us to do this immediately, for
from these equations we find (putting % = 0) that the ve-
locities parallel to the axes of x and y of any particle in the
plane of any are
daa
dy
-= «,,,*;
and hence the whole resolved velocity in the plane of xy of
that particle, viz.
will be o)3 <\/a? + y2.
Now if we suppose the particle to be at a distance unity
from the origin, and therefore a?2 + y2 = 1, this velocity will
become a>3, and hence w3 is the resolved velocity in the plane
of xy of any particle situated in that plane, at a distance
9
66
unity from the origin. Since the particle is rigidly con-
nected with the origin, it is evident that this velocity takes
place perpendicularly to the line joining the particle and the
origin ; and also since — is negative, it is evident that this
dt
velocity tends to move the particle from the axis of x to-
wards that of y,
In the same way it may be proved that MI and <o2 are
similar velocities with respect to the planes of yz and zx,
respectively ; o>i tending from the axis of y towards that of
#, and o>2 tending from the axis of % towards that of sc.
71- To give clear ideas, we shall represent the manner
in which these velocities tend, by means of the following
figure.
Let XY, FZ, ZX be the intersections of the planes
of acy, yz, »#, respectively, at the time #, with a sphere
fixed in space, unity being its radius, and the origin its
center ; then <a3 will be the resolved velocity along the great
circle XY of any point P situated on that great circle, and
o>j will be the resolved velocity along the great circle FZ
of any point Q situated on that great circle, and o>2 will
be the resolved velocity along the great circle ZX of any
point R on that great circle; and these velocities tend in
the directions represented by the arrows.
67
It is evident from this, that wj, o>2> «>3 are also the
angular velocities of the planes of yz, ##, and ay, round
the axes of #, y, and *, respectively : but it may be easily
seen from the equations (.6), that wlt w25 <Ws are not tne
angular velocities round the axes of #, y, ar, of any other
points of the body, except those situated in the respective
co-ordinate planes.
72. We may determine the position and motion of the
body by means of these velocities, as follows.
Let A be any fixed point on the surface of the sphere
described before (see the figure), and AB any fixed great
circle; draw the great circle AZC to meet YX produced
in C.
Let the angle ABZ = \^, the angle A Z = 0, and the
angle C Z X = <p; then it is evident that these angles define
completely the position of the body in space at the time t,
and may be considered as the co-ordinates of the position
of the body, \^ and 0 being the co-ordinates of the point Z,
which we shall call the pole of the body, and (p the addi-
tional co-ordinate requisite to determine the position of the
plane of ZX, and therefore of the whole body; we may
easily determine these co-ordinates from the quantities
«!, w2, fc>3, as follows.
It is evident that the velocities of the point of the body
coinciding with Z are,
, perpendicular to A Z,
o/ 1
J f\
and — along ^Z,
and the velocity along CX of the point of the body co-
inciding with C is
d\l/
--| sin AC, due to the variation of >Jr,
+ -T"5*0 ZC*» due to the variation of 0;
68
i. e., since ZC = 90°,
Now by what we have proved in (71), the velocities of
the body coinciding with Z are,
&>2 along ZJT,
and - wl along Z F,
and the velocity along C X of the point coinciding with C
is o>3 ; hence, since these two sets of velocities must be
equivalent, we have, resolving the latter set so as to make
them coincide with the former,
-J- sin 9 = — MI cos 0 + o>2 sin <
at
— = ojj sin 0 + o)2 cos 0
(C).
73. These differential equations connect 0, \^, and 6
with «i, &)2» o>3, and these, along with the three others (A')
in (69), which connect o^, w2, w3 with the impressed forces,
form a system of six equations connecting the six unknown
quantities o)15 w2, ft>3, \^, ^, 0 with #, and which therefore,
when it is possible to do so, will enable us to solve any
problem respecting the motion of a rigid body about its
centre of gravity.
The equations (B) will enable us to determine the mo-
tion of any point we please of the body, should it be
requisite to do so.
74. These equations will be sufficient for our present
purpose, namely, the determination of the Earth's motion
about its centre of gravity ; we shall hereafter recur to
this subject, and deduce several interesting consequences
from these equations.
CHAPTER VI.
PRECESSION AND NUTATION.
75. WE shall now make use of the equations deduced
in the last Chapter, to determine the motion of the Earth
round its center of gravity.
The forces which act on the Earth, are the attractions
of the Sun, Moon, and other planetary bodies; but on
account of the Earth's nearly spherical form, the motions of
the Earth round its center of gravity, produced by these
forces, are but small ; hence, by the principle of the super-
position of small motions, we may consider, separately and
by itself, the effect of the attraction of each planetary body
on the motion of the Earth round its center of gravity ; we
shall accordingly commence with the Sun's effect.
76. We shall first prove, that the attractions of the
Sun on any particle of the Earth are the same very nearly
as if the Sun's mass were condensed into his center of
gravity.
Take the center of gravity of the Sun as origin; let
xy'% be the co-ordinates of any element §m of it, and let
xyz be the co-ordinates of the attracted particle §m of the
Earth. 9
Then, if V denote the sum of each element of the Sun,
divided by its distance from $m, the attractions of the Sun
on $m will be
dV dV dV
dx" dy* d%'
70
\
Now V = 2 —
2 (o?^ + yy + #*') r'
putting r and r' for <#2 + y2 4- #2, and X* 4- y'2 + #'2, re-
spectively
a?<j?' + yy + %% \
r }
expanding and neglecting the squares, &c. of the very small
x' 11 %' , r
quantities — , — , — , and -
r r r r
1
= — 2owi .
r
Since 2^m,t?', 25my', 2^m^' are zero, because the origin is
center of gravity.
Hence F= — ;
r
and therefore, neglecting the squares &c. of very small
dV dV dV
quantities, F, and consequently the attractions — , — , — ,
are the same as if the whole mass of the Sun were collected
into its center of gravity.
77- Now take the principal axes of the Earth, passing
through its center of gravity, as the axes of co-ordinates,
the polar axis being that of % ; let aoy% be the co-ordinates
of any element $m of the Earth, and xyss the co-ordi-
nates of the Sun, supposed to be condensed into his center
of gravity ; then, m being the mass of the Sun, his at-
tractions on £m, parallel to the axes of x and y, will be
andF=
71
and hence the quantity N, (see the equations (A') of the
last Chapter) which equals 2$w (Yx — Xy)^ will become
(y'-y)*-(*'-«>)y \
*'-*m^)2+(*'-*)i* '"I
\yoe-aiy) [r'2-2(a!X+yy + **') + r2]"! \ ,
r and r being the distances of $m and $m from the origin,
and r being therefore very large compared with a?, y\ %9
or r ; hence, expanding and neglecting the squares of very
small quantities, we have
or, since the origin is center of gravity, and therefore
2^ma?, S^my, 23m*, each zero, and since the axes are
principal axes, and therefore
each zero, we have
and ^f being the same as in the last Chapter.
Similarly,
8m' *V
- Q,
r
Sm'y'ss'
(C-B).
r'5
Hence the equations (-4')j in the last Chapter, become
A + (C - B) W2»3 = (C - K)
72
78. These equations simplify very much in the case
of the Earth, for the polar axis being that of *, the
moments of inertia round all axes in the plain of xy will
be the same, since the Earth is symmetrical with respect
to the polar axis ; hence B = A, and the last equation will
become
and therefore o>3 = constant = n suppose.
That is to say, by (71), the angular velocity of the plane
of the equator round the fixed axis with which the polar
axis coincides at any time, is a constant quantity.
Now if ri be the Sun's mean angular velocity relative to
the Earth,
n'2= — — , m being the Earth's mass,
= — , very nearly, m being very small com-
pared with TO';
C — A
hence, and putting ft for — - — , the other two equations
become
79- Now in the figure (page 66), let S be the point
where a line drawn from the center of the Earth to the Sun
meets the fixed sphere mentioned in page 66; draw the
great circle SZ, then SZ will be the Sun's north polar dis-
tance, which we shall denote by A, and SZX will be the
Sun's hour-angle relative to the meridian plane ZX, which
we shall denote by h : it is evident that A and h are the
polar co-ordinates of the Sun, and therefore we have
73
x = r sin A cos A,
y = r sin A sin A,
z' = r cos A.
hence -r- = sin A cos A sin A,
ra
— '— = sin A cos A cos h ;
r a
and hence our equations become
— + w/3 wa = 3w'2/3 sin A cos A sin A,
2 - w/3o>i= — 3n'2fi sin A cos A cos A.
ct£
We shall take a year as the unit of time, and hence n',
2ir
which = - — , will become 27r, and n will be about 365,
a year
also we shall shew that /3 is about = — .
330
Now h varies in consequence of the Earth's diurnal
rotation, and also in consequence of the Sun's motion and
the motion of the polar axis ; but the part of its variation
due to the former cause is very much greater than that due
to the latter causes, hence we may put
dh
where $n is a small quantity compared with n, depending
on the motions of the Sun and polar axis, we put - n because
h decreases with the time.
Hence in the first of our equations, putting
d<a\ dh)\ dh dto, , *
it becomes
da,, nfi 3n'*fi
e»>2 = — — s— sin A cos A sin n ;
dh - n 4- $n - n +
10
74
or, neglecting $n compared with w,
— - - /3ft>2= — Sra'* — sin A cos A sin h.
ah n
Now the second member of this equation is very much
smaller than the first, on account of being multiplied by
- , hence in it we may, when integrating, consider the
n
periodical quantity A as invariable, since it varies very
slowly compared with h ; and therefore putting, for brevity,
3n'2ft .
— sin A cos A = y,
n
our equation becomes
^ -£«,•• -y sin h... (1),
where y is a very small quantity which may be considered
invariable in integrating.
In the same way we shall have
-~- +
ah
80. Now differentiating (l), and adding (2) multiplied
by ft to it, we have
_J2
-jj£ + ft2u,} = (- y + 7/3) cos h
= - y cos h,
neglecting ft compared with unity.
The integral of this equation will be
o>i = A cos (fih + B) + C cos h,
where A and B are arbitrary constants, and C a constant to
be determined by substitution. Now A and B, since they
depend only on the initial circumstances of the motion, are
independent of the Sun's action ; and, as it is our object to
75
determine the effect of that action alone, we shall omit the
term
Acos(fih+ B),
which does not depend on it, and we shall have simply
(Dl = C cos h,
so far as the Sun's action is concerned ; substituting this
value in the equation, we find
C(-i + /r> = -7,
or C = y, neglecting ft2 ; hence
fcjj = y cos h.
In like manner we shall have, differentiating the equa-
tion (2), and subtracting (1) multiplied by /3 from it,
d2a)2 „„ .
— — + /52ft>2 = - 7 sin h ;
an
and therefore, as before,
ft>2 = y sin h.
81. Having thus determined &>! and ft>25 we shall sub-
stitute their values in the equations (C) (last Chapter), in
order to determine 9 and 0, and so find the position and
motion of the pole ; we have then, substituting the values of
o>i and ft>2 just obtained in the two first of the equations (C),
dvl/
— ~ sin 9 = y sin h sin 0 - y cos h cos <p
= - y cos (h + 0),
dO
and — = y sin h cos (f> + y cos h sin 0
= y sin (h + 0).
Now in the figure (page 66), the angle SZX is A, and
the angle XZC is 0, hence the angle *S*ZC is (f) + h; there-
fore if we take the point A to be the pole of the ecliptic
(which we may do since its position is arbitrary), it is evi-
dent that ^ + h or SZC will be the Sun's right ascension
76
— 90°; hence, if a be the Sun's right ascension, we shall
have
$ + h = a - 90°,
and hence our equations become
d^ .
— — sin 9 = — y sin a,
dO
- = -7cosa;
or, putting for y its value,
d^ . 3n'2Q .
— - sin 6 = — sin A cos A sin a,
at n
d9 3riz$ .
— = sin A cos A cos a.
dt n
82. Now if / be the Sun's longitude, /, a, and 90° - A
are the sides of a right-angled spherical triangle, the right
angle being opposite Z, and 0 (the obliquity of the ecliptic)
being the angle opposite 90° — A ; hence since, by Napier's
rules,
cos A = sin 6 sin I (1),
cos Z = sin A cos a (2),
and sin a = cot A cot 9 (3),
we have
sin A cos A sin a = cos2 A cot 0, by (3),
= sin 0 cos 9 sin2 Z, by (l),
and sin A cos A cos a = cos Z cos A, by (2),
= sin 9 sin I cos Z, by (l) ;
hence our equations become
<ty 3n'20
I_ = r cos 0 sm2 /,
dt n
d9 3n'*P .
a- _ _ L- SJn 0 Cos /.
dt n
77
83. Now — being very small, we may in integrating
n
these equations consider & invariable in the terms multiplied
by — ; and we may also, for the same reason, consider I to
n dl
vary uniformly, and therefore put — = n ; hence, putting
d\/ d\, dl d
d9 dO ,
Tt=-din>
and sin2 / = 1 (l - cos 2 Z),
our equations become
S— £*
hence, integrating
^ + C = - - - y3 . cos 6 (I - J sin 2/),
Q + C" = 3 - 3 sin 0 sin /.
n ^
In the second members of these equations 0 may be
considered as the mean obliquity of the ecliptic, and may
therefore be determined by astronomical observations; and
— , the ratio of a year to a day, may be similarly deter-
n
mined ; and thus we may put our equations in the form
9 + C = gfi sin /,
where e', f ', g', are numerical quantities got from obser-
vation.
These equations determine the effect of the Sun's at-
traction on the motion of the Earth round its center of
gravity.
78
84. To determine the effect of the Moon on the Earth's
motion round its center of gravity, we may proceed in ex-
actly a similar way, merely supposing the symbols which
before referred to the Sun now refer to the Moon, and alter-
ing them accordingly, as follows :
Let my, w/5 and rt be the mass, mean angular velocity,
and distance of the Moon, then as in Art. (78), we shall
have
here m instead of being much smaller than m t as before, is
much larger than it, and therefore cannot be neglected
as before,
let X = — , then
n
m
,
and therefore — ' = - — ;
r, 3 X + 1
hence, in changing our equations so as to refer to the Moon,
M*
we must in substituting for n2 put — '— instead of nf
for it.
Hence, if we put dashes under the letters to denote
that they refer to the Moon, we shall have for the Moon's
effect on the Earth's motion round its center of gravity,
s ' / - sin '"'
Of course the point A in the figure, (page 66), is now
supposed to be the pole of the Moon's orbit, and not the
pole of the ecliptic.
79
85. In these expressions the coefficients are much
smaller than before; so small that the periodical quantities
multiplied by them, viz. sin 2ntt and sin nt^ which go
through all their values in half a month, and a month
respectively, may be neglected, this will give
and
and therefore --- = - - •—- — cos 9 ,
dt 2n (X + 1)
, dB
and — ' = 0,
dt
Which equations prove, that the effect of the Moon's
attraction (omitting very small periodical quantities of short
period) is to produce a motion of the pole of the Earth per-
pendicular to the great circle AZ, i.e., the great circle
joining the pole of the Earth and the pole of the Moon's
orbit ; and the velocity with which this motion takes place
(i.e. sin 0,), is
cos 9 sin
We shall resolve this velocity along and perpendicular
to the great circle, joining the pole of the Earth and the
pole of the ecliptic, in order to get our quantities measured
in the same way as before in the case of the Sun, and so de-
termine the variations (due to the Moon's action), of the
angles \js and 0, which refer to the pole of the ecliptic.
86. Let i be the inclination of the Moon's orbit to the
plane of the ecliptic ; then it is evident that t, 0, and 9t form
the sides of a spherical triangle.
80
Let cr be the angle made by 6 and 0/? then resolving the
velocity
3«2/3
— cos 9 sm 6 ,
2w (X + 1)
(which acts perpendicularly to 0y), along and perpendicular
to 9 we find for the resolved parts,
-- 7-* - - cos 9 sin 0 cos a perpendicular to 0,
2n (\ + 1)
and -- -r-* - - cos 9f sin 9t sin cr along 0,
2 w (A. + 1}
and hence, since — sin 9 and — are these velocities, we
at at
have the following equations to determine the effect of the
Moon's action on \!/ and 9, viz.
d\js 3nffl cos Qt sin 9f cos cr
1/7 2n(\+ 1) sin 0
d0 3w//3
— = -- — cos 9 sin 0 sin cr.
dt
Now in the triangle of which t, 0, and 0/ are the sides,
o- is the angle opposite i, and if & be the longitude of the
Moon's node, it is evident that & is the angle opposite Bt ;
hence we have
cos t = cos 9 cos 0/ + sin 0 sin 0y cos cr, ... (l)
cos Qt = cos i cos 0 + sin i sin 0 cos & , ... (2)
sin 0. sin t
(3)
am 56 alu O"
(•n« rt sin A r-i\< rr
hence
sin0
COS t - COS 0 COS 0,
= (cosi cosfl + siru sin0cos& )
81
cos t sin2 0 — sin t cos0 sin 9 cos &
sin2 9
(sin20-cos20)
cos2i cos 0+sm t cos i — — — - - - cos & -sm2t cos 9 cos2 &
sin 9
— : — —
sin 9
and also
cos 9, sin 9 1 sin a-
= (cos i cos 9 + sin t sin 9 cos & ) sin t sin £
= ^ sin 2t cos 9 sin & + ^ sin8 1 sin 0 sin 2 & .
In these expressions Q> alone may be considered as va-
riable, all the other quantities varying very slowly, and
within very small limits; also, since these expressions are
to be multiplied by a very small coefficient, we may, in the
periodical terms, neglect sin2t, since t is not much more
than 5°.
Also, if v be the mean angular retrograde velocity of
the Moon's nodes, we may put
and hence we shall have as in former cases
3w2/3 A sin2* 1
' cos0{cos-< ----- sin 2 1 cot 29 cos Q },
i/(\+l) 1 2 j
cos0sm2t sin & ;
— - --
d& 2ni/(\
d9
and hence, integrating,
9 + C' = - - — / ^ ^ cos0 sin2t cos & ;
11
82
or, as before, we may put these equations in the form
3 3
^tc-^ — «-/
A + 1
When e^ ft, and g are numerical quantities, got by ob-
servation.
Thus we have determined the effect of the Moon's
attraction on the motion of the Earth, round its center
of gravity.
87- The effects of the other planetary bodies are very
small indeed, and we shall neglect them ; hence, adding
together the effects of the Sun and Moon, we find for the
whole motion of the Earth round its center of gravity, so
far as it is affected by external attractions,
>/, + C = - /3 (el - ~^~ Q ) + /'/3 sin 2 /
A. + 1
0+ C' = 's\i
/
A + I
The first term of the expression for >// + C is non-
periodical, and its rate of variation is
or -(3(e'n' + ^-_
which being constant and negative represents a uniform
retrograde motion of the pole in longitude : it gives rise to
what is called the precession of the equinoxes, because, in
consequence of it, the first point of Aries moves constantly
backwards, and therefore the equinox occurs sooner than it
otherwise would every year.
83
Observation shews that this retrograde motion of the
pole in longitude is about 50".l per year; hence we ought
to have
The other terras of \|/ + C and 0 + C' are periodical,
depending on the longitude of the Sun and of the Moon's
nodes; they are called the solar and lunar nutations. Ob-
servation shews that the coefficient of sin & is about 18",
and that of cos Q about 9".5, the coefficients of the other
terms are much smaller ; hence we ought to have
and 9».6-«r,r-- ............ (3).
A + 1
88. Since e , //5 and g are known numerical quan-
tities, it is evident, that from (l) combined with (2) or (3),
we may eliminate )3 and find X ; the result is
X = about 70,
thus by observation on precession and nutation we may
determine X, which is the ratio of the Earth's mass to that
of the Moon.
By the same equations we may determine /3, the re-
sult is
/3 = about .00319, or - .
*oO
89- Now /3 may be also calculated by integration, if
we know the law of arrangement of the Earth's mass ; we
shall calculate /3, assuming the results arrived at in Chap, in,
and if we find that the value of /3 thus obtained coincides
with that just determined by observation, it is evident that
we shall have an additional proof of the correctness of our
hypotheses in Chap. in.
84
90. We have evidently
»-») 2Sm (*»-«»)
' S1
#, y, # being the co-ordinates of any particle ($m) of the
Earth, the polar axis being the axis of ss ; or using the polar
co-ordinates as before,
Now putting ^ + ^ cos 20 for cos20, and integrating
relatively to 0, observing that r does not contain 0, the
numerator of )3 becomes
which, putting
r'=«5{l +5e(J -!.")},
and observing the property of Laplace^s coefficients in Art.
21, becomes
9 9 -
8
~ 45'
hence our integral becomes
Sir ra d(o5e) .
— / p -- da.
15 Jn
da
85
In like manner the denominator of /3 becomes
TT f r«, d(O/4 /i \1
= - / / p— : — { -- -- M V
5 J-\ Jo r da (3 \3 ) I
Since, on account of the smallness of /3, we may neglect e
in its denominator.
Hence, observing the property in Art. 21, the denomi
nator of /3 becomes
STT /•<•! rf(a5) STT /•«.
— / p — — aa which = — / pa*da.
15 Jo * da 3 A ^
hence we have
5
/ p 1
1 <J Q (t (t
r^
I pa*da
«/o
Now the equation for determining e in Art. 53, gives
e, /•"' . 1 ra' d(a5e) a>zaj
-- ^ / pa'da + — - / p— — da + -- ^ = 0,
a,J0 r *«/•'• ^a STT
r"> w^o/
a e / pa^da -- '-
0 ' Vt r STT
hence p = -
rpa* da
. '
a~ I pa? da
' Jo '
I pa*da
putting
m being, as before, the ratio of the centrifugal force to
gravity at the equator. We can go no farther in calculating
/3 without knowing the law of density ; hence, taking the
law already assumed, and substituting in the integrals and
performing the integrations, we shall find
/3 = about .0031359, or .
330
91. Hence this value of /3 coincides with that got from
observation, in Art. 88, and we have therefore an additional
proof of the hypothesis of the Earth's fluidity, or rather,
of the assumed law of density ; for since this result cannot
be obtained without assuming the law of density, it is not of
much value in proving the hypothesis of original fluidity ;
but we may consider that hypothesis as well established by
previous results, and then the coincidence of the values of /3
will go to proving the probability of the assumed law of
density.
END OF PART I.
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A 000426824 9
Sh'n'tlii n-ilt hs PuUixlicd. !>;! the wwr Aitthnr :
MATHEMATICAL TRACTS,
PART II.
Containing, among other subjects, The MATHEMATICAL
THEORY of ELECTRICITY, briefly considered,— The
LIBRATIONS of the MOON, ^- The THEORY of
JUPITER'S SATELLITES; and a Short ESSAY on
TIDES.
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